Average Error: 45.6 → 0.0
Time: 1.3m
Precision: 64
Internal Precision: 384
\[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 2092550.302196164:\\ \;\;\;\;\frac{\frac{i}{2}}{\sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0} \cdot \sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0}} \cdot \frac{\frac{i}{2}}{\sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}\\ \end{array}\]

Error

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 2092550.302196164

    1. Initial program 43.9

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(i + i\right) \cdot \left(i + i\right) - 1.0}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\frac{i}{2} \cdot \frac{i}{2}}{\color{blue}{\left(\sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0} \cdot \sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0}\right) \cdot \sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0}}}\]

    5. Applied times-frac0.1

      \[\leadsto \color{blue}{\frac{\frac{i}{2}}{\sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0} \cdot \sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0}} \cdot \frac{\frac{i}{2}}{\sqrt[3]{\left(i + i\right) \cdot \left(i + i\right) - 1.0}}}\]

    if 2092550.302196164 < i

    1. Initial program 47.3

      \[\frac{\frac{\left(i \cdot i\right) \cdot \left(i \cdot i\right)}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right)}}{\left(2 \cdot i\right) \cdot \left(2 \cdot i\right) - 1.0}\]

    2. Applied simplify31.2

      \[\leadsto \color{blue}{\frac{\frac{i}{2} \cdot \frac{i}{2}}{\left(i + i\right) \cdot \left(i + i\right) - 1.0}}\]

    3. Taylor expanded around inf 0

      \[\leadsto \color{blue}{0.00390625 \cdot \frac{1}{{i}^{4}} + \left(\frac{1}{16} + 0.015625 \cdot \frac{1}{{i}^{2}}\right)}\]

    4. Applied simplify0

      \[\leadsto \color{blue}{\left(\frac{0.00390625}{{i}^{4}} + \frac{\frac{0.015625}{i}}{i}\right) + \frac{1}{16}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(FPCore (i)
  :name "Octave 3.8, jcobi/4, as called"
  :pre (and (> i 0))
  (/ (/ (* (* i i) (* i i)) (* (* 2 i) (* 2 i))) (- (* (* 2 i) (* 2 i)) 1.0)))